Multistability resulting from random rearrangements in a ring
resonator with a nonlinear element
V. V. Zverev and B. Y. Rubinshtein
Opt. Spectrosc. (USSR) 62(4), April 1987
Abstract
Certain types of random rearrangements of attracting sets (attractors) arising in the
phase space of a ring resonator containing a nonlinear medium are studied numerically. It is
shown that for certain values of the parameters we can reduce the 2-D mapping describing
the evolution of the field in the resonator to 1-D and give a qualitative explanation of the
threshold phenomena observed. A method is considered for expansion in terms of a small
parameter that allows us to calculate the fine structure of a random attractor analytically.
As it has been shown in [1] - [5], self-oscillations transformed into random motion (optical
turbulence) can arise in a ring resonator with a nonlinear element (henceforth to be called the
nonlinear ring resonator – NRR). This paper is devoted to the study of random rearrangements
of attractors in the phase space of the NRR (including random-regular motion transitions), leading to behavior similar to hysteresis. Such behavior was observed in the numerical modeling
of NRR dynamics in the adiabatic regime (the Ikeda mode [1]); it was assumed that nonlinear
conversion of light takes place due to the ν-photon transitions in the two-level medium. It
is important that at the time of rearrangements of an attractor, an exchange (jump) of the
type of NRR dynamics takes place; such a transition can result both in the smooth variation
of some parameter of the system and in impulse perturbation of the exciting field. This indicates, in principle, feasibility of the design of new types of multistable (having a series of
stable states) optical devices based on NRR (we call multistable dynamic systems or states, to
which correspond before and after the jump isolated attracting sets, but stationary points are
optional).
We consider a theoretical NRR model. Let a two-level medium placed in the ring resonator
be excited by an external light source (stationary and coherent) through a beam splitter; after
passing through the active medium part of the light leaves the resonator through another beam
splitter; the intensity reflectance of each of the mirrors is equal to κ.
In the case of ν-photon transitions, the slowly varying electric field amplitude E of the
wave, the polarization P of the medium, and the density W of population inversion satisfy the
equations
E 0 + c−1 Ė
Ṗ
Ẇ
= iβgP ∗ (E ∗ )ν−1 ,
P
= − + i(ω0 − νω)P + ig(E ∗ )ν W,
T2
1+W
= −
+ i(g ∗ E ν P − c.c.)/2,
T1
(1)
(2)
(3)
where c is the speed of light; T1 is the energy relaxation time; T2 is the dephasing time; ω0 is the
resonant frequency of the medium; ω is the frequency of the exciting field; g and β are constants
1
characterizing the medium; derivatives with respect to z and time t are denoted by prime and
dot, respectively. For T2 ¿ Tr , where Tr is resonator round trip time, the NRR dynamics (in
the traveling-wave regime) can be described by a set of differential-difference equations
Ψ(t) + 1
Ψ̇(t) = −
+ (2βl0 )−1 | E(t − Tr ) |2 [1 − exp(2χν (t))],
T1
√
E(t) =
1 − κ Eex + κE(t − Tr ) exp[(1 − iδ)χν (t) + iθ],
(4)
(5)
where
χ1 (t) = βl0 | g |2 Ψ(t)T2 /(1 + δ 2 ),
(
χν (t) = −
1
2(ν − 1)βl0 | g
ln 1 −
2(ν − 1)
1 + δ2
Ψ(t) =
1
l0
Z l0
0
|2
T2
)
|E(t − Tr )|2ν−2 Ψ(t) , ν > 1;
W (t − Tr + z/c, z)dz,
(6a)
(6b)
(7)
Tr = l/c, δ = (ω0 − νω)T2 , θ = (ω − ωc )Tr , l0 is length of the active region of the NRR, l is
the total resonator length, ωc is the frequency of the isolated mode of the field, Eex and E(t)
are complex amplitudes of the electric field of the exciting wave and wave at the entrance to
the active medium, respectively. We omit a detailed description of the procedure for deriving
Eqs.(4)-(6b), discussed in [3] for ν = 2 and is easily generalized to the case of arbitrary ν. By
assuming additionally that βl0 |g|2 T2 /(1 + δ 2 )|E|2ν−2 Ψ ¿ 1, δ À 1, T1 ¿ Tr , we can describe
the dynamics of the NRR by 2-D mapping
"
EN +1
#
iθ + iR|EN |2ν+2
,
= Eex + κEN exp
1 + |EN |2ν
(8)
p
√
where EN = αE(N Tr ), Eex = α(1 − κ)Eex , R = βl0 δα/T1 , α = (|g|2 T1 T2 /(1 + δ 2 ))1/ν . We
imply using Eq. (8) that the amplitude of the field at the entrance to the active medium changes
abruptly at t = N Tr , where N = 1, 2, 3, . . ., and remains constant during each time interval
N Tr < t < (N + 1)Tr ; thus, the phase trajectory of the NRR (in discrete time scale) is obtained
as a result of iteration of mapping (8). In this paper, we will consider only the dynamics of
mapping.
We note that for ν = 1, |EN | ¿ 1, a simpler mapping is derived from Eq. (8)
³
´
EN +1 = Eex + κEN exp i(θ − R) − iR|EN |2 ,
(9)
it was studied in [1, 2]. By representing the results of iteration of mapping (8) by points in
the complex E plane, we can observe different types of NRR dynamics: convergence of the
sequence of points to a limiting (stationary) point; convergence to each j-th group of points
(j = 1, 2, . . . , H), including points with numbers N = j + HK, K = 0, 1, 2, . . ., to its limit
(cycle of multiplicity H); random motion. In the last case the points form an image of random
(strange) attractor (SA), normally having the form of a thick curve or a set of curves. By
considering an enlarged fragment of such curve, we can discover that it consists of several close
quasiparallel curves (first-order fine structure); each of these curves in turn possesses a similar
second-order structure, and so on. Thus, the SA of mapping (8) is sturturally similar to the
Henon attractor described in [7], and is the direct product of a 1-D manifold and the Cantor
set (also see [1] and [3]). For certain combinations of the values of the NRR parameters, we can
analyze the SA structure by reducing the 2-D mapping (8) to approximate 1-D mapping. The
2
method of reduction [8] is a modification of the method used earlier in [6] for SA of polynomial
mappings. Introducing polar coordinates EN − Eex = rN exp(iΦN ), we rewrite Eq. (8) in the
form
rN +1 = κ|Eex + rN exp(iΦN )|,
ΦN +1 = ΦIN +1 + ΦII
N +1 + θ,
(10)
where
ΦIN +1 = arg(Eex + rN exp(iΦN )),
ΦII
N +1 = G(rN +1 /κ),
G(w) =
Rw2ν+2
.
1 + w2ν
(11)
We may assume Eex is real. We note that it follows from Eq. (10) that rN +1 ≤ κ(Eex +rN ) (the
triangular inequality) due to the fact that rN ≤ rM = κEex (1 − κ), where rM is a stationary
point dominating the mapping rN +1 = κ(Eex + rN ); so that the sequence {rN } is finite. Using
this, we can show that for κ ¿ 1 the following estimates are valid
|ΦIN | ≤ rM /Eex ∼ κ,
|ΦII
N | ∼ G(Eex ),
(12)
and, if furthermore, Eex ∼ R ∼ 1, then |ΦIN | ¿ |ΦII
N |; this inequality can be satisfied also for
other combinations of the parameter values. By taking ΦIN to be small, we introduce the formal
expansion parameter ΦIN → ²ΦIN and seek an equation of SA as the equation of a family of
invariant curves of mapping (10), by representing the r.s.h. of this equation in the form of a
series in ²
r = S0 (Φ) + ²S1 (Φ) + . . .
(13)
(after completing the calculations we should set ² = 1). In our case
S0 (Φ) = κG−1 (Φ − θ),
κS0 (γ(Φ)) sin γ(Φ)
κ
arctan
,
S1 (Φ) = − 0 −1
G (G (Φ − θ))
Eex + κS0 (γ(Φ)) cos γ(Φ)
(14)
(15)
where G−1 (.) is a function inverse to the function defined in (11); γ(Φ) is a multivalued function
defined by the equation
Φ = G(|Eex + κG−1 (γ − θ)eiγ |) + θ.
(16)
Retaining only S0 in Eq. (13) which corresponds to replacement of ΦIN in Eq. (10) by
zero, we rewrite Eq. (13) in the form of a parametric equation of the curve of least detailed
representation of SA, neglecting fine structure of any order
√
√
E = Eex + κ v exp(iG( v) + iθ).
(17)
The NRR dynamics in such an approximation is determined by a 1-D mapping
√
√
2
vN +1 = Eex
+ 2κEex vN cos(G( vN ) + θ) + κ2 vN
(18)
2 /κ2 (the zeroth approximation for mapping (9) was
of curve (17) onto itself; here vN = rN
considered in Ref. [9]).
Retaining the terms S0 and S1 in Eq. (13) (linearization in ΦIN ) we describe first-order fine
structure. An enlarged segment of SA of mapping (9) for Eex = 6.73, κ = 0.15, R = θ = −1
3
Figure 1: Fine structure of random attractor.
−2 for these data) is shown in Fig. 1. The points correspond to results of nu(|ΦIN /ΦII
N | ∼ 10
merical iterations; the initial point was chosen in curve (17) for v = 11π [in the case of mapping
(9), we should use G(w) = R(1 − w2 ) in (17)]. The theoretical graph of SA in the zeroth (first)
approximation is shown by solid (light) circles and agrees well with numerical results. We note
that calculation is required of all orders in expansion (13) for exhaustive analytical description
of the Cantor structure of SA; the resulting function is infinite-valued. Rearrangements of the
attractors and corresponding threshold phenomena discussed in this paper are due to the existence of multicomponent attractors; each of components of such attractor has its own basin
of attraction. We can observe in the numerical experiments both merging of the individual
components taking place with variation of the controlling parameter and jumping of the phase
trajectory from one component to the other, caused by impulse perturbations of the exciting
wave. Using the approximate mapping (18) we illustrate how multicomponent attractor can
emerge. A graph of the function (18) for ν = 2, Eex = 1.05, κ = 0.33, R = 40 and θ = 2.5
is shown in Fig. 2. We can reproduce the iteration process graphically by drawing from any
point of the curve horizontally to the intersection with the bisector of the angle between the
axes, and from the intersection point found, vertically to the intersection with the mapping
graph. We can see that the selection of the initial point in regions I, II or III leads to locking of
the trajectory in the corresponding region; the trajectory never leaves this region (within the
region, motion can be both regular and random). With arbitrary initial point, the trajectory
is sooner or later will be trapped in one of these regions. Thus, in this case the attractor has
three components. The condition of trapping can be violated by varying any of the parameters
determining the shape of the curve (18) (fragments of the mapping graph are shown in the inlay
of Fig. 2 for two situations: A–no trapping, B–trapping takes place). Changing the control
parameter to lift the trapping condition, and then returning to the initial value, we can cause
transition to another component. For |ΦIN /ΦII
N | ¿ 1, mappings (8) (exact 2-D) and (18) (approximate 1-D) have multicomponent attractors for close values of the parameters. Hence, it is
convenient in the search of such attractors to begin study of the 1-D mapping that shows the
appropriate combinations of parameters, and only after this, proceed to the numerical iteration
4
Figure 2: Approximate 1-D mapping (18) in the case of two-photon transitions.
of mapping (8). Same results obtained this way are tabulated. Mapping (8) was iterated for
ν = 2, Eex = 1, R = 18, and different κ and θ). Each group of three symbols, containing (not
containing) the point, characterizes a two-component (three-component) attractor; the point
replaced the symbol of the component, for which the locking condition ceases to be satisfied
(phase trajectory is locked by the adjacent component). In order to characterize the dynamics within one component, we use the notation: S means stationary point; 2, 4, 8 stand for
cycles of corresponding multiplicity; X – randomness; X – randomness with a double-cycle
structure (the latter means that the group of points with even and odd numbers have close
and distant regions of localization from one another, but, on the whole, the motion is random).
θ
3.375
3.350
3.325
XXS
XX.
XX.
4X.
4X.
2X.
2X.
XXS
XXS
XX.
4X.
4X.
2X.
2X.
.XS
XXS
XXS
8XS
4X.
2X.
2X.
.XS
XXS
XXS
XXS
4XS
4X.
4X.
.XS
.XS
XXS
XXS
XXS
4XS
4X.
.XS
.XS
.XS
XXS
XXS
8XS
8XS
|
0.56
.XS
.XS
.XS
XXS
XXS
XXS
4XS
.XS
.XS
.XS
.XS
XXS
XXS
XXS
|
0.565
.XS
.XS
.XS
.XS
.XS
XXS
XXS
.XS
.XS
.XS
.XS
.XS
XXS
X.S
.XS
.XS
.XS
.XS
.XS
.XS
X.S
.XS
.XS
.XS
.XS
.XS
.XS
.XS
κ
Numerical modeling of the dynamics of mapping often leads to results hardly amenable to interpretation. It is difficult to distinguish the complex periodic motion from the random or one
type of randomness to another. In this situation, it is convenient to use additional information,
including dependence of the maximum Lyapunov index (MLI) [10] on some system parameter.
The MLI characterizes the rate of divergence (convergence) of close phase trajectories, averaged along a certain trajectory, and can be found numerically; the MLI is positive for random
5
Figure 3: Dependence of maximum Lyapunov index on exciting field amplitude.
motion, and negative for stationary point or limiting cycle. In our case, calculation of the MLI
allows us to obtain detailed information on variation of the structure of the attractor in the rearrangement region. Dependence of the MLI L on Eex , for mapping (9) (R = θ = −1, κ = 0.15)
is shown in Fig. 3. A two-component attractor exists in the regions I-IX; variations corresponding to the various components are shown by points and circles. We observe for each of
them transition of randomness according to the Feigenbaum scenario [11] the Eex values for
which cycles of multiplicities 2 and 4 and randomness arise are denoted by numerals V, VI,
VII (circles) and VIII, IV, III (points). The dependence of MLI for a one-component attractor
is shown by points to the left of I and right of IX, The transition between stationary point
and randomness is denoted by the numeral X, while the window of stability in the random
zone by numeral II. We can assume that threshold phenomena and multistability are due to
rearrangement of attractors, and this is one of the typical scenarios of behavior of a nonlinear
system with delay (elementary scenarios of onset of randomness are given in Ref. [12]). A
crisis randomness described in Ref. [13] is associated with this scenario. Study of the means
of altering SA and search for scenarios is of definite interest in connection with prospects of
the design of optical logic devices and computer memory cells. We note that rearrangements
of attractors in systems with continuous time and in the distributed systems were discussed in
Ref. [14]. Some other threshold effects in systems with SA leading to typical dependence of
MLI on the parameter are considered in Ref. [15].
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6
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